Probab. Theory Relat. Fields 99, 171-195 (1994) Probability …georgii/papers/LDPsuper... · 2008....

Probab. Theory Relat. Fields 99, 171-195 (1994) Probability Theory a.d Related Fields

�9 Springer-Verlag 1994

Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction

Hans-Otto Georgii

Mathematisches Institut der Universit~it Mtinchen, Theresienstrasse 39, D-80333 Miinchen, Germany

Received: 2 June 1993/In revised form: 3 January 1994

Summary. For Gibbsian systems of particles in R a, we investigate large deviations of the translation invariant empirical fields in increasing boxes. The particle interaction is given by a superstable, regular pair potential. The large deviation principle is established for systems with free or periodic boundary conditions and, under a stronger stability hypothesis on the potential, for systems with tempered boundary conditions, and for tempered (infinite-volume) Gibbs measures. As a by-product we obtain the Gibbs variational formula for the pressure. We also prove the asymptotic equivalence of microcanonical and grand canonical Gibbs distributions and establish a variational expression for the thermodynamic entropy density.

Mathematics Subject Classification (1991)." 60F10, 60G55, 60K35, 82B05, 82B21

1 Introduction

One of the classical themes of Equilibrium Statistical Mechanics is the study of the fluctuations of extensive quantities, such as the particle numbers and the energies of particle configurations in finite boxes, in the infinite-volume limit. This includes the problems of existence and variational characterization of the pressure, and of the asymptotic equivalence of the Gibbs ensembles on the level of measures. (A detailed discussion of these problems and their physical background can be found in the lectures of Lanford [11] and Martin-Lrf [12].) For spin systems on a lattice, all these questions have been reconsidered successfully in the light of the recent progress in large deviation theory, see [8] and the references therein. In the case of continuous systems of point particles in Euclidean space, the situation is less satisfactory; the status so far is essentially still that set out by Lanford [11]. In this paper we develop a large deviation theory for such particle systems, with the aim of contributing to a systematic study of the questions above.

The general setting is as follows. We consider the Euclidean space R d of any dimension d > 1. A configuration of particles (without multiple occupancies) is de-

172 H.-O. Georgii

scribed by a locally finite subset co of R d, i.e., a set co C R d having finite inter- section with every bounded set. We write f2 for the set of all such configurations co. ~2 is equipped with the a-algebra 9 c generated by the counting variables Ne:co--+ card(co N B) for Bore1 subsets B of R d. It is well-known [10, 13] that 5 c is the Borel a-algebra for a natural Polish topology on Q. The translation group 0 = (OX)xERd acting on ( ~ , S ) is defined by 0xco = {y - x :y E co},co c f~,x c R a. The mapping (x, co) -+ 0xco is known to be measurable [13]. We let 7 9 denote the set of all probability measures P on (g?,Y) with finite expected particle numbers P(N~) - fNAdP in bounded Borel sets A C R d, and we write 7)o for the set of all O-invariant P E 79. For each P E 790 there exists a number p(P) < oc, the intensity of P, such that P(NA) = p(P)IAI for all Borel sets A. Here IAI is the Lebesgue measure of A.

We introduce a topology rE on 7) as follows. Let /2 denote the class of all measurable functions f : f2 -+ R which are local, in that f(co) = f ( co n A) for some bounded Borel set A and all co E ~2, and tame, in that Ill <c(1 +NA) for (without loss the same) A and some constant c < oc. The topology zL of local convergence is then defined as the weak* topology on 79 relative to /2, i.e., as the smallest topology on 7 ) making the mappings P --+ P ( f ) =_ f f d P continuous. In particular, the mappings P -+ P(NA) for bounded Borel sets A are continuous relative to zL. This shows that zc is much finer than the usual weak topology on 7) which is based on the above-mentioned Polish topology on t2.

The basic objects of interest for studying large deviations are the O-invariant empirical fields in increasing boxes. ( For general information on concepts, use, and recent developments in large deviation theory we refer to the monographs [1, 2].) Thus, for each n > 0 we consider the half-open cube An = [ - n - �89 + �89 of

volume Vn = (2n + 1) d and the associated translation invariant empirical field

R,,~o = v~l fA 6Ox~O(,)dx (1.1)

of any configuration co E (2. In (1.1), we replaced co by the An-periodic continuation

co(n) : { x + ( 2 n + 1)i :x E coNAn, i E Z d}

of its restriction to An. This has the advantage of making Rn,~ translation invariant. Thus R,,o~ E 79o for all n and co. I f 7)o is equipped with the evaluation a-algebra generated by the maps P --+ P(A),A E U, the random measure R, : co ~ Rn,~ be- comes a measurable mapping from ~ to 79e. The asymptotic behavior of the empirical fields can be described as follows: For each P E 79o,

z~ - lim Rn = Pz in P-probability, (1.2) n ---~ o o

where co ~ P~ is a regular conditional probability of P relative to the a-algebra 2- of O-invariant sets in 5 r. This follows immediately from Wiener's multidimensional mean ergodic theorem, cf. [10].

In this paper we study large deviations from the ergodic theorem (1.2) when P is Gibbsian relative to a suitable pair interaction q~. More precisely, we establish a large deviation principle for Rn when the particles are distributed according to a Gibbs distribution in An with free or periodic boundary condition and the underlying potential cr is superstable and satisfies a decay condition called regularity (see Theorem 2). Under the stronger hypothesis that tp diverges at the origin sufficiently fast, we obtain a uniform large deviation principle relative to Gibbs

Large deviations for Gibbsian particle systems 173

distributions with boundary configurations ( E ~ satisfying a uniform condition of temperedness (Theorem 3(a)). By virtue of the well-known superstability estimates of Ruelle [18], this leads to a large deviation principle for tempered Gibbs measures on (f2,5 c) (Theorem 3(b)). The rate function is, of course, given by the excess of the free energy density over its equilibrium value. In particular, we establish the Gibbs variational formula which asserts that this equilibrium value is given by the pressure (Eqs. (2.15) and (2.22)). A basic ingredient of all this is the existence and lower semieontinuity of the (internal) energy density (Theorem 1). Finally, we prove a limit theorem for conditional Poisson distributions of microcanonical type, implying the equivalence of Gibbs ensembles on the level of measures (Theorem 4). This is an instance of the maximum entropy principle and is closely related to a (microcanonical) Gibbs variational formula for the thermodynamic entropy density.

In the Poissonian case of no interaction, the analoguous results were obtained earlier in [10]. Our results here rely heavily on this paper. A weak version of a large deviation principle for particles with superstable interaction of finite range is also contained in [15]. The case of particles with hard core (which is contained in the present work) was already treated in [9].

2 Statement of results

We begin describing the particle interactions which we will consider. We assume, for simplicity, that the interaction is only pairwise and thus given by an even measurable fimction (p : R e --+ R U {oc}. Such a 9 is called a potential For each n> 0 ,

1 Hn(o9) =-- Hn,free(Co) = ~ Z (O(V - x), ~ E ~, (2.1)

x,yCrofqAn,x @y

is called the associated Hamiltonian in An with free boundary condition. A potential (O is said to be stable if there exists a constant b = b((o) < ec such that

Hn > - bNn for all n _>- 0. (2.2)

Here Nn = NA,. In particular, (2.2) implies that (p > - 2b. Sufficient conditions for (O to be stable can be found in [17] . Let us say that (O is purely repulsive if (O is nounegative and bounded away from zero near the origin, i.e., if there exists some 3 = 3((O) > 0 such that

(O>~1s (2.3)

Here and below, I.I stands for the maximum norm on R d. A potential (O is called superstable if

(O = (os + (O,- (2.4)

for a stable (O~ and a purely repulsive (o r. The use of this concept was revealed by the pioneering work of Ruelle [16, 18] (Related ideas appeared also in the independent work [3, 4].)

Besides the hypothesis of superstability which assures that large particle numbers in a bounded region require a large amount of energy we shall also need a condition on the decay of (o. A potential (o is called lower regular if there exists a decreasing function q/: [0, eel--+ [0, c~[ such that

(o(x) > - O(Ix[) for all x E R a (2.5)

174 H.-O. Georgii

and OO

f~ ( s )J - lds < co. (2.6) 0

q0 will be called regular if q) is lower regular and, in addition, there exists some r(q~) < cc such that

~o(x)<~(Ix]) whenever I x] >r((p). (2.7)

Our first result is the existence of the energy density o f any P E 79o. To state it we need to recall that the Palm measure of P E 79o is defined as the unique finite measure po on (f2, 5 c) satisfying

f P(d03) ~ f (x, 0x03) = f dx f P~ (x, 03) (2.8) xEo9

for all measurable functions f : R d x (2 ---+ [0, oo[. We have po((2) = po({03 E (2 : o3 ~ 0}) = p(P), and the normalized Palm measure p(p)-lpo can be viewed as the natural version of the conditional probability P('1{03 E (2 �9 03 ~ 0}); see [13] for more details. Let us introduce the set

79~)) = {P E 790 "P(N 2) < oo for all bounded Borel sets A} (2.9)

of all second-order elements of 790. For each n > 0 and 7:' E 790 we let

r = v21p(H,), (2.10)

be the expected cp-energy per volume in An. In view of (2.2), 4~n is well-defined (possibly equal to +oo), and ~bn > - bp. In Section 3 we shall prove the following.

Theorem 1 Suppose ~o is superstable and lower regular. Then, for each P E 790, the limit 4~(P) = l imn+ ~bn(P ) exists and satisfies

po(fe) if P E 79~) ~(P) (2.11) l oc otherwise '

where 1

f~(03) = ~ Z q~(Y)' 03 E (2. (2.12) 04=yEo)

Moreover, the function ~ : 7)o -+ R U {oo} is lower semicontinuous (relative to zE) .

4~(P) is called the energy density of P. It is clear that 4~ is affine. To state our large deviation results we next need to introduce the entropy density. We let Q E 79o denote the Poisson point random field on R d with intensity p(Q) = 1. For each P E 790 and n > 0 we write

Pn = P({03 E (2 : (gM An E "})

for the restriction o f P to An. We think of Pn as an element of 7 ~ which is supported on f2n = {03 E (2 : 03 C An}. The negative entropy density of P is then defined as the (existing) limit

I(P) = lim v2~I(Pn; Qn), (2.13) n----+ o c

Large deviations for Gibbsian partMe systems 175

where 1(.;. ) stands for the relative entropy; see [10] for more details. I is an affine function with ~L-compact level sets [10].

Suppose now we are given an inverse temperature/3 > 0 and an activity z > 0. (Later on, we shall assume without loss that the units are chosen so that/3 --- z = 1.) The excess free energy density of P E 790 is then given by

Iz,~(P) = I(P) +/3~(P) - p(P)logz + p(z,/3), (2.14)

where p(z,/3) = -min[ I + flt.b - p logz]. (2.15)

Theorem 2 below asserts that p(z,/3) is nothing other than (/3 times) the pressure, and Iz,~ is the rate function in a large deviation principle for the distribution of the empirical fields Rn under the Gibbs distributions with free or periodic boundary conditions. (Configurational boundary conditions will be considered later in Theorem 3.) For n > 0, the Hamiltonian in An with periodic boundary condition is defined by

Hn,per(Og) = vn~(Rn,~o) 1

= ~ ~ ~0(y-x). (2.16) xGcofqAn,y~co(n),y+ x

The last equation follows from (2.11) and the easily verified fact that the Palm measure of Rn,o~ is given by

R~,~ = Vn 1 ~ 60xo)(,~, (2.17) xG~oNAn

cf. [10]. For a given boundary condition be E {per, free}, the associated Gibbs distribution in An with parameters z,/3 > 0 is defined by

Pnr = Z~-clLbczU"(~ (2.18)

where Zn,z,/~,be = Qn (zN"exp[--/3Hn,be]) (2.19)

is the so-called partition function. It follows from (2.2) resp. Theorem 1 and (3.3) below that Zn,~,#,bc is finite. Thus Png,~,bc is well-defined. Again, we think of it as an element of 79 with support f2,.

Theorem 2 Let ~o be superstable and regular, z, fl > O, and F : 79o ~ R U { ~ } a measurable functional satisfying F > - e(1 + p) for some c < co. Then, for bc = per or free,

lim sup v~-alog Pn,z&be (e -v"F(R")) <--_ -- inf[I~,~ + F~sc] (2.20) ?/----+ OO

and lim infv;l logPng~ue(e , _>- - inf[Iz,/~ +FUSe], (2.21)

where Flsc is the largest lower semicontinuous minorant and F use the lowest upper semieontinuous majorant o f F relative to rL. In addition, Iz,~ has vL-compaet level sets, and

176 H.-O. Georgii

p(z, fl) = nlimoo v~- 1 log Z n , z , f l , b c . (2.22)

Note that (2.20) and (2.21) take the familiar form of a large deviation principle when F is chosen to be zero on some measurable set A C 7>o and equal to +oc outside A. The existence of the limit in (2.22) is the classical result on the existence of the pressure; p. 68 of [17] contains the relevant bibliographical notes. The coincidence of the right sides of (2.15) and (2.22) is called the Gibbs variational formula. (In the case of a hard-core interaction, this variational formula has already been proved in [6].) The upper bound (2.20) will be proved in Sect. 4 and the lower bound (2.21) in Sect. 5.

We now turn to large deviations for Gibbs distributions with configurational boundary conditions, and for infinite-volume Gibbs measures. We need some nota- tions. Let C = [-1/2, 1/2[a= A0 be the centered half-open unit cube and L = Z a. The sets C + i , i E L, form a partition of R d. For n > 0 we set Ln = LM An = (i ~ L : Ii I <n} and

N 2 Tn = Z c+i. (2.23) iELn

For t > 0 we define

~(t) = {Tn<tvn for all n>0}. (2.24)

The configurations in ~2" = Ut>0 fa(t) are called tempered. The multidimensional

ergodic theorem shows that P(~2*) = 1 for all P E 7>~). For each ~ E Q* and n > 0 we let

Hn,~(oo) = Hn(r + ~ q)(y - x) (2.25) xCooNAn,yE~\An

denote the Hamiltonian in An with boundary condition ~. The associated Gibbs distributions Pn~,/~,~ are defined by (2.18) with bc = ft. By Lemma 4.2, the last sum in (2.25) exists when p is regular. Under the hypotheses of Theorem 3 below, Lemma 6.1 and the estimates in the proof of Lemrna 6.3 even imply that Zna,~g < cx), so that Pna,/~,( is well-defined. A measure P E 7 > is called a tempered Gibbs measure for z, fl > 0 if P(~?*) = 1 and, for all n > 0 and measurable functions f > 0 on O,

P( f ) = fP(d~)fPn~,~g(do~)f(oJ U (~ \ An)).

Note that the identity above is equivalent to the equilibrium equations in [18]. Let us say q0 is non-integrably divergent at the origin if there exists a decreasing

function ~( :]0, oc[---+ [0, oc[ such that

and

~o(x)>z([x[) whenever q0(x)>0 (2.26)

f z(s) s d- ~ ds = oc. (2.27) 0

Together with the lower regularity, this condition implies that (p is superstable, see Proposition 3.2.8 of [17]. It also follows that tempered Gibbs measures exist; this was proved independently in [3, 18]. Under this hypothesis, the following theorem


provides a uniform large deviation principle for Gibbs distributions with (uniformly) tempered boundary conditions, and a large deviation principle for tempered Gibbs measures. Its proof will be given in Sect. 6.

Theorem 3 Suppose ~o is regular and non-integrably divergent at the origin. Also, let F be as in Theorem 2 and z, fi > O. (a) For each t > O, we have

lira sup v~-llog sup Pn,z,~g(e -v"F(R")) <= -inf[Iz,/~ +Flsc] (2.28) n---+oc ~Ef2(t)

and lira infv~-llog inf Pn,z~(e -v~F(Rn)) > - inf [ /~ + FUSe]. (2.29)

n ~ o c ~E~?(t) . . . . = '

Moreover, (2.22) holds with bc = ( uniformly for all ~ ~ f2(t). (b) Each tempered Gibbs measure P with parameters z, B satisfies a large deviation principle for Rn with rate function Iz,~, in that inequalities (2.20) and (2.21) hoM with P instead o f Pn~&bo.

Let us note that an application of the contraction principle to Theorems 2 and 3 leads to analoguous large deviation principles for the individual empirical fields R~, in (2.17); see [10] for more details. We also note that, under the hypotheses of Theorem 3, the rate function I~,~ vanishes precisely on the set of all O-invariant tempered Gibbs measures with parameters z,/~. One direction of this variational principle follows by standard arguments from Eq. (I.2) and the upper bound in Theorem 3 (b). The reverse direction can be obtained by an adaptation of the proof of the analogous result in the lattice case, see Theorem (15.37) in [7]. Details will be provided elsewhere.

Our last result is a version of the equivalence of ensembles. For any nondegenerate interval D C [0, oc[ and real e we consider the microcanonical Gibbs distribution

QnlD,c,per ~ Qn(" INn E vnD, On, per ~ VnC) = Qn('lp(Rn) E D , ~ ( R n ) < r (2.30)

in An with periodic boundary condition. As we will see, the conditioning event has positive probability for all sufficiently large n whenever c > inf~b(D). Here q5 : [0, oc[---, R U {oc} is defined by

~b(v) = inf{~b(P) : P c 7~o,p(P) = v,I(P) < oc}. (2.31)

Since ~b, p and 1 are affine, ~b is convex, q5 is finite on an interval [0, v(q))[, where v(cp) = oc except when ~o has a hard core, see Lemma 7.1. Intuitively, v(~o) is the close-packing density of {q) = ec}-balls.

We write aCCn~ooP (nl for the set of all accumulation points (relative to ~ ) of a sequence P(") in 7 ~.

Theorem 4 Suppose qo is superstable and regular, let D C [0,o c[ be a nondegenerate interval with in fD < v(q O, and e > infqS(D). Then there exists some fl > O and z > O such that

0 + a c e QnlD,e,per C {Iz,p = 0 } D a c c Pn,z,/Lper + (~" ( 2 . 3 2 ) n ----+ o o n - - + o o

178 H.-O. Georgii

In particular, / f {Iz,~ = 0} consists of a unique element P~,B then

n--+o~lim QnlD,e,per : n--+cclim Pna,fl,per = Pz,&

Moreover,

lim v;llogQ,(Nn E vnD, Hn,per <= Vne)

= - i n f { l ( P ) : P E 7~o, p(P) E D, ~(P)<r (2.33)

I f e < inf~b(D) or infD > v(r the limit in (2.33) equals - ~ .

(2.32) expresses the asymptotic equivalence of microcanonical and grandcanon- ical Gibbs distributions. (2.33) is an analogue of the classical existence result for the thermodynomic entropy density, cf. [11, 17]. The difference here is that the particle number is not fixed but ranges in a whole interval, and that we use periodic boundary conditions. In addition, (2.33) provides a microcanonical Gibbs variational formula. The proof of Theorem 4 will be given in Sect. 7 which also contains some additional information on the involved functions. With the same techniques, one can derive the asymptotic equivalence of small canonical and grand canonical ensembles. In addition, an obvious extension of Lemma 7.2 in the spirit of Sects. 4 and 5 leads to a large deviation principle for the empirical fields R, under the microcanonical distributions QnlD,e,per. We leave this to the reader.

3 The energy density

In this section we prove Theorem 1. We begin with some basic estimates. For P E 79o we set p(2)(p) = p(N2), where again C = Ao is the centered half-open unit cube. Since

m m

N Z < m ~ U ] j when A c U A j , (3.1) j=l j=l

p(2)(p) < ~ if and only if P E 79~ ), cf. (2.9). Suppose now we are given a superstable lower regular potential ~o. The presence

of the purely repulsive part q~r of ~o implies that, for fixed n, Hn tends to infinity quadratically as Nn ~ ec. This is asserted in our first lemma. Recall the definitions (2.23) and (2.10) of Tn and ~n.

Lemma 3.1 There exist constants a > O, b < cx~ such that for each n>=O

Hn >aTn - bN, (3.2)

and ~n >=aP (2) - b p. (3.3)

Proof (3.2) follows from (2.4), (2.2) and (2.3) by partitioning An into k a vn half- open cubes Aj of size 1/k (where k is the smallest integer exceeding 1/6 for the number 6 in (2.3)), and applying (3.1) with A = C + i,i E Ln, and m = k d. (One can take a = fi/2k a. b exceeds the constant in (2.2) by 6/2.) (3.3) follows from (3.2) by integration. I


As a consequence of (3.3), if P r ) then e n ( P ) = oc for all n > 0 and thus

l i m n ~ 4~n(P) = oe. To prove Theorem 1 we can therefore assume that P E 7)~). In this case, q~,(P) admits a convenient description in terms of the Palm measure P~ of P.

Lemma 3.2 For all P E 7)~)) and n>O we have ~n(P) = P~ where

1 f~,n(co) = ~ ~ qo(y)(& n ( & - y)llv,, co e Q.

04:yE~o

Proof Consider the (measurable) function

1 fn(X, CO)----5 Z go(y)lA, n(A,-y)(X),

04yEm

x C Rd,~o E f2. Since ~o>__ - 2b, f~(x,~o)> - blA,(X)Nn(O_x~O). Also,

f P~ f dxl A,(X)Nn(O-x~) = P(N 2) <

for all P E 7)~ ), by (2.8). Equation (2.8) therefore also holds for f =f~. But for f =f~, the left-hand side of (2.8) coincides with r and the right-hand side with P~

Since [An N (An -y)I/vn ~ 1 as n ~ ~ , it is natural to expect that P~ --+ P~ which will give Us the first assertion of Theorem 1. To make this rigorous we need the lower regularity of ~p. Let again L = Z a and, for each i E L,

Oi = O(d(C, C + i)), (3.4)

where d(C, C + i) = ([i1 - 1)+ is the distance of C and C + i. Hypothesis (2.6) implies that ~iEL Oi < C~. Moreover,

1 f~ >= - 2 Z Oi Nc+i, (3.5)

iEL

and the function on the right-hand side is P~ for all P E 7)(0 2) because for all i E L

P~ = fP(dc9) Z Nc+i(Oxeo) x@o)f')C

<=P(NcNA1 +i) <=P(N~)I/2p(N] 1 )1/2

< 3a p(2)(P). (3.6)

In the last step we used (3.1). It follows that (b(P) = P~ is well-defined for all

P E 7)~). We now compare ~(P) with r

Lemma 3.3 There exists a sequence e, -+ 0 such that for all P E 7)(~) and n>O,

4)(P) >= q~n(P) - enP(2)(P).

180 H.-O. Georgii

Proof Let P E 7)(o 2) and n > 0 be given. Since IAn M(An - y ) l / vn< l , (2.5) and Lemma 3.2 imply that ~(P)__> ~ n ( P ) - en(P), where

n(n) : SP~ }2 * )tAn \ (An O * y E c o

Distinguishing the cubes C + i containing y, we conclude from (3.4) and (3.6) that ~n(P) <= enP(2)(P), where

en = ~ 3 d ~ i m a x l A n \ ( A n - y ) l / V n . yCC+i

iCL

But the dominated convergence theorem shows that ~n ---+ 0 as n -+ oc.

Proof of Theorem 1 We first show that, for all P E 7)0, limn~oo ~n(P) exists and

has the claimed value. The case P ~ 7)(o a) was already discussed after Lemma 3.1.

For P E 7)~), Lemma 3.2 and Fatou's lemma imply that

~b(P) = P(lim inf fe,n) <l i ra inf ~n(P) n---+ oo n---+ oo

because the functions f~,n are not less than the right-hand side of (3.5). Together with Lemma 3.3, this shows that ~b(P) = l i m n ~ ~n(P). To prove the lower semicontinuity of ~b we note first that each ~bn is lower semicontinuous. This is because Hn satisfies (2.2) and is thus the supremum of functions in Z;. Now let c E R and (P~)~cD be a net in { ~ < c } which converges (in z~) to some P E 7)o. Then p(P~)-+ p(P) < oo. We thus can assume without loss that s _-- sup~ p(P~) < Pc. In view of (3.3) and the first part o f this proof, we have for all ~ E D

e > ~b(P~) >ap(Z)(P~) - bp(P~)

and thus p(Z)(p~)< (c + bs)/a ==- C. Together with Lemma 3.3 this implies that

Cbn(P) - erie'< lim inf ~bn(P~) - enc' c~CD

< lira inf ~b(P~)__< c e C D

for all n > 0 . Letting n --+ oc we see that P E {~__<c}.

We conclude this section proving the compactness of the level sets of the functionals I~,/~ defined in (2.14).

Lemma 3.4 For any two numbers cl, c2, the set {I + q~ < cl + c2 p} is zc-compact.

Proof The set above is closed because p is continuous and I and ~b are lower semicontinuous. In fact, I even has compact level sets, see Proposition 2.6 of [10]. The same is true for P , the relative entropy density with reference measure QZ, the Poisson point random field with intensity p(QZ) = z > 0. It is easy to see that I z = Iz,0 = I - plogz + z - 1. By (2.2), (b > - bp. The set under consideration is therefore contained in the compact set {P < cl + z - 1 }, where z = exp(b + c2). ,


4 The upper est imate

Let ~o be superstable and regular and F �9 7~o --+ R U {oo} a measurable function such that F > - c(1 + p) for some c < oc. In this section we shall prove the following result.

Proposit ion 4.1 For bc = per or free,

lim sup v21 logQ~(exp[-v,F(Rn) - H,,bc]) < -- inf[I + �9 +Flsc]. t / - + OG

In the case bc = per, the above proposition follows directly from the results of [10] combined with those of Sect. 3. Namely, let G = F + ~. The integrand under consideration then equals exp[-v,G(R,)]. By Theorem 1 and (3.3), G > - c ' ( 1 + p) with c ~ = c + b, and G is clearly measurable because so is ~, Theorem 3.1 of [10] thus implies that the lira sup in Proposition 4.1 is not larger than - inf[I + Gist]. But Glsc >F l sc§ ~ because �9 is lower semicontinuous.

The proof of Proposition 4.1 for bc = free is based on a comparison with the case of periodic boundary conditions. We need several lemmas. First, we use the regularity of (p to estimate the interaction of suitably separated configurations. For each integer k > 0 we define

1 6k = ~ ~ c~O(~')v<, (4.1)

g_>-k

where OO(f)= O ( f - 1 ) - O ( f ) > 0 for f > l , 0 O ( 0 ) = 0, and ~ is as in (2.5) and (2.7). By (3.4),

0i ---- ~ 0O(f) (4.2)

<_->t~1

for all i E L. Hence

C>~O iCL

and thus 6k --+ 0 as k --~ oc. For all n > 0, co E f~n and ( E f2 we have

~(y - x) <_ ~ ~',-jXc+i(co)Nc+j((). (4.3) xEeo, yE~ iELn,jEL

We estimate the long-distance contribution to the right-hand side of (4.3) in two cases. (The second case will be used later in Sect. 6.)

L e m m a 4.2 For all n > O, co ~ f~, and k >__ O,

Z ~li-j Nc+i(co)Nc+j(~) iELn,jEL:Ii--j[ >=k

< ( 6 k ( l + 2 a ) T . ( c o ) if i f=co (m) for some m > n, = ~ 6k[Tn(co)+v.t2 d] i f ( E f 2 ( t ) for s o m e t > 0.

Proof. In view of (4.2), the sum above is not larger than

182

g > k iCLn,jEL E +i

Using the inequality uv < (u 2 + v2)/2 we obtain the upper bound

1

g > k

with S~,l(~) = Z Nc+j(~) 2 card (L, N +j)) .

j@L

For ~ = co (m) with m > n we have

H.-O. Georgii

Sn/(CO(m)) = ~ NC+/(co)2 Z card {]" C Le + i : j =--j' mod 2m + 1) j l CLn iELn

< Vn(2dv~/vm)Tn(co) <2dv~Tn(co).

On the other hand, if ~ E (2(t) then

__< (v, A

VnAflV2(nW) <= t2d v~vl

which implies the temma in the second case.

Recall the notation r(cp) in (2.7).

Lennna 4.3 For all n>=O and k >r(~o), [Hn - Hn+k, pe~[ <2dc~kTn on f2,.

Proof After a comparison of (2.1) and (2.16), the lemma follows immediately from (2.5) and (2.7) together with (4.3) and Lemma 4.2. !

Next we need to compare R~ with R~+k. For n > 0,s > 0 we define

f2(s,n) = {co ~ f~ " T~(co)<sv,}. (4.4)

Lemma 4.4 For all k > l , s > 0 and f C ~,

lim sup IR~+k,o)(f) -- Rn,o)(f)l = O. n--+~ )

Proof The case of boundedf is trivial. We thus assume for simplicity that [fl <N~ for some centered cube A D C w i t h f = f ( . A A). For each n we can write, setting m = n + k ,

]R,~,~o(f) - Rn,~o(f)] <(v21 - vml) f A, N~+x(co(m))dx

+ v2 t fAm\A, NA+x(co(m))dx

where 0An = {x ~ An : x + A ~ An}. On the other hand, for each m with Am D A + C and all VCAm we have


fvXa+x(co(m))dx< ~Nc+,(co(m))r{x ~ V : (A + x) n (C + i) +{3}1 iEL

<2alAI E Nc+i(~ icLA(V+A+C)

<4C/IAI E Nc+j(oa) jELm(V)

<4alAI card Lm(V)l/2Tm(~) 1/2

with Lm(V) : {j ELm :j -- i rood 2m + 1 for some i E V + A + C}. In the cases V : Am, V : Am \ An, and V = (3A, we have card Lm(V)<-_ca[V I for some constant CA < oo. Combining all estimates above one can now easily complete the p roof

Lemma 4.5 For any three constants c1,r 3 > 0 there exists some s : - s ( c 1 , 0 2 , c 3 )

< oo such that Qn (eC'N"-czr"; Q(s, n) c) _--< e -e3v"

for all n>O.

Proof Let s be so large that sc2 > e3 + e c~ - 1. The result then follows from the inequality

1 a(~,.)o < exp[c2(T, - svn)]

and the fact that Q,(e clx") : exp[(Wl - 1)v,].

Before completing the proof o f Proposition 4.1 we need one further notation. We let U denote the system of all sets of the form

{ " maxlPl ( f i ) -P2( f ) '<=~} U : (P1,P2) E 7"90 X 'PO l<-i<k

with k > 1, e > 0, and ]] . . . . . J~ E s By definition, H is a uniformity base for z ; .

Proof o f Proposition 4.1 in the case be = free. For given s > 0 and arbitrary n we can write

Q, (exp[-v,F(R,) - Hn]) = p,(s) + q,(s),

where p,(s) is defined by restricting the integral on the left-hand side to the set O(s,n) and q,(s) is the remaining contribution corresponding to O(s,n)C Let z > 0 be any given number, and let s = s(c + b, a, z + c) be chosen according to Lemma 4.5. Here c is the constant appearing in the hypothesis F > - c ( 1 + p), and a,b are as in (3.2). It then follows that qn(s)<exp[-zv,] for all n. To estimate the main term p,(s) we choose an arbitrary integer k>rOp ). Then for each n > 0 we can write, setting again m = n + k and using Lemma 4.3 and (2.16),

pn(s) : eVm-V"Qm(exp[-vnF(Rn) - Hn]; f2(s,n) 71 {NAm\A, = 0})

< exp[Vm -- vn + 2a6ksv,]Qm(exp[-Gm]),

where Gm: vnF(R~) + Vm~(Rm) § oo. ]O(s,m)C. Next we choose any U E H and f > 0 and d e f i n e F l : F A f ,

F[s(P) : inf{Fz(P') : (P,P ') E U},

184 H.-O. Georgii

and G~: = F ~ + ~b. Lemma 4.4 asserts that (Rm,Rn) E U on O(s,m) for sufficiently large m. Since vnF > vmF t - (Vm - vn)& we conclude that for these m

Vm 1Gm > G~(R~) - (1 - Vn/Vm)(.

But this is all what is needed for the large deviation upper bound (3.2) o f [10], cf. the proof of Lemma 5.6 there. (Measurability of G~ is not required.) Hence

lira sup v~llog Qm(exp[-am]) < - inf [ I + (G~)ls~]. n---+ ~

In view of the lower semicontinuity of 4~, (G~j)lso >_- - (F(z)ls~ + 4. Also, Lemma 3.4 shows that the argument of Remark 1.4 of [8] can be applied, yielding

Hence

sup inf [ I + �9 + (F~)lsc] = inf [ I + q~ +Flsc] --- 7. [>O,UEIA

lim sup v~- l log pn(S) <= 2dbkS -- 7" t/---+ OO

Since k is arbitrary, we finally get

lira sup v~-llog[p~(s) + q~(s)] __< - 7 A ~, n - - + o o

and letting z ~ oo we obtain the result.

5 The lower estimate

We still assume that q) is superstable and regular. Our proof of the lower bound (2.21) follows the standard device of changing the measure so that untypical events become typical, and controlling the Radon-Nikodym density by means of McMillan 's theorem. But some refinements are necessary. The basic observation is that the familiar approximation of invariant by ergodic probability measures can be sharpened as follows. For q > 0 we define

Fq = {co E (2 : (p(x - y ) < q and [x - y [ > l / q for any two distinct x,y E co}. (5.1)

It is easy to check that Fq is measurable.

L e m m a 5,1 Let P E Po be such that I(P) + ~b(P) < oo. For each open neigh- bourhood U o f P and any e > 0 there exists some O-ergodic U c U such that I(P') < I(P) + ~, 4)(U) < (b(P) + 8, and U(Fq) = 1 for some 0 < q < ec.

Proof 1) Let n > 0 be given. Since ~(P) < oo, we have p(2)(p) < ec and thus, by Lemma 3.3, ~n(P) < oc. Hence Pn(Fq) T 1 as q T co. Therefore we can choose a number 0 < q(n) < oc such that Pn ~ Pn(Fq(n)) - - + 1 as n ---+ oo. We also fix an integer k>__ 1 such that 2k>r(~0), where r(~0) is as in (2.7). We can assume without loss that q(n)>-O(k) for all n. In the following we use again the abbreviation m = n + k .

2) Let /3(n)E ~P be the probability measure relative to which the particle configurations in the disjoint blocks Am § (2m + 1)i,i C L, are independent with


identical distribution Pin =- Pn('[Fq(n)) �9 This means in particular that the corridors (Am \ An) + (2m + 1)i, i E L, contain no particles. We also set

p(n) = v~l fA,j,(n) o 021dx.

It is then obvious that p(n) E 7Jo, and a standard argument shows that p(n) is ergodic;

see, for example, Theorem 14.12 of [7]. Since IFq(n) is O-invariant and P(n)(Fq(n)) = 1, it also follows that P(n)(Fq(n) ) = 1.

3) Next we show that lim SUPn__+~I(P(n) ) <I(P). By an analogue of Lemma 5.5 of [10],

I (P (n)) ~ l (Ptn; Qm)/Vrn

= [I(P,(.fq(n)); Qn) + Vm -- Vn]/Vrn.

On the other hand, since I(P) < oe we have Pn << Qn with a density fn, and thus

I(Pn(. [Fq(n)); Qn) = Pn(10g f~/p,; Fq(n))/p, <(I(Pn; Qn) + 1)/pn -- log Pn

because xlog x > - 1 for all x > 0 and thus

Pn(log f~; Fq(n)) = Qn(f~log fn; Fq(n)) ~ - 1.

Since (Vm - V,)/Vm --+ 0 and Pn ~ 1 as n -+ ee, the desired result follows. 4) The regularity of q~ implies that lira SUPn_,~(P(n))<O(P ). Indeed, since

obviously p(n) E 5~ ), (2.11) and (2.8) yield

j)(p(n)) = f P(n)(&O) ~ f~(Oxe)) xC~oNC

v~fAmdUfP(n'(d~ E 1c+,(x)~o(y-x) x,yEo),y4,x

= Vm lf/6(n)(d~ ~ x, yCo3,y@x

= an + bn.

~o(y - x ) l a m n (x - C)l

Here an = vmlP(n)(Hn)~ and

bn = l)mlfp(n)(do)) 1 E

xE~oMAn,yCo)\ Arn ~o(y - x),

and we have used that for P(n)-almost all (9 and all x E co either x E An, and thus x - C C A m , or Am A (x - C) = 0. Now,

an = Pn(Hn; Fq(n))/pnVm ~n(P)vn/p.Vm + bP(Nn; rcq(n))/pnVm

_-< ~n(P)(1 + 0(1)) + bP(N2.)~/2(1 -pn)l/2/pnVm = ~b.(P)(1 + o(1)) + o(1)

186 H.-O. Georgii

because of (2.2) and (3.1), and b~ = o(1) because of (2.7), (4.3), the inequality

= / 3(n) ~N, "/3(n)'N , < ~p,2, 2 P(n)(Nc+iNc+j) t c+i) t c+j)=Pt ) /Pn

for i E Ln,j ~ L~, and the summability of i --+ r 5) We finally show that P(")( f ) ---+ P ( f ) as n --+ oo for all f E s Since this is

quite obvious when f is bounded, we assume for simplicity that If[< N~, where A is any cube w i t h f = f ( . N A). Then we can write

]e(n)(f) _ r ( f ) [ =< (sn § tn)/Vm

sn = f {x+ACAm}dXlP,(f 00~lFq(~)) - P ( f o Ox)l

and

tn ~ faAmdx ~(n)(N~+x) + P(NA+x)] ,

where •Am is as in the proof of Lemma 4.4. Now P(NA+x) = [Alp(P),

13(n) (NA+x) = Pn(NA+xmod2m+l IFq(.)) < IA Ip(P)/P.,

and therefore t, = o(vm). On the other hand, the integrand defining sn is at most

11 -- Pnl lPn(N~+x; Fq(n)) + Pn(NA+x ; Fq(,))

_-< [A [p(P)[ 1 - pn~[ + P(N])I/z(1 - pn) V2,

whence s, = O(Vm). This completes the last step of the proof.

The main advantage of the classes

790,q = {P E 790 " P(Fq) = 1} (5.2)

is the following continuity property of ~b.

L e m m a 5.2 For each q > O, 4) is continuous on 79O,q.

Proo f For each n >0, Hn is local and bounded on I'q. Hence ~bn is continuous on 79O,q. Also, a glance at Lemma 3.3 and its proof shows that the convergence ~b~ --~ ~b is uniform on Po,q. This gives the result.

Let r>r(cp) be a fixed integer. For n>-_r we define the modified empirical fields

R#~,~ = R,,o~C~A,_r. (5.3)

L e m m a 5.3 For each ergodic P E 7)o, R#n ---+ P in P-probability as n --~ oo.

Proo f This follows in the same way as (1.2); cf. the proof of Remark 2.4 in [10].

Let F be as in Theorem 2.

Proposit ion 5.4 For all P E 790 and bc E {per, free},

lim inf v2 t logQ~(exp[-v,F(Rn) - H~,bo]) > -- [I(P) + r + FUSe(P)]. R----+ O ~

with


Proof We may assume without loss that the right-hand side of the asserted inequality, denoted by -TP, is finite. By Lemma 5.1, we can even assume that P is ergodic and supported on F = Fq for some q > ~b(r). For n > r let Pn # = Pn-r. We think of Pn # as a measure on f2n which leaves Dn = An \ An-~ free of particles. Since I(P) < oc,Pn-r << Qn-r with a densityf~_~. Hence P~# << Qn with the density f f = l{ND=O}fn-~exp[vn- vn-~]. Given any e > 0, we define

An = {F(Rn) < FUS~(P) + e, vn~H~,bc < ~(P) + ~, v;l logff < I(P) + e}.

A well-known estimate (see, e.g.,[10]) then shows that the lira inf in the proposition is not less than

- 7 p - 3e + lira inf v~ -1 log P~(An).#

It is therefore sufficient to show that # P~(An) ~ 1. But

P (F(Rn) < Fu (P) + = P(F(R ) < Fu ~ + 1

because of Lemma 5.3, and

P#~( v211og f~ < I(P) + e) = Pn_~(v~-llog f~_~ + 1 - v21vn_~ < I(P) + e) ~ 1

by McMillan 's theorem [5, 14]. Moreover, for bc = per we obtain from Lemmas 5.2 and 5.3 that

Pn # (vn-lgn,per < ~(P) + ~) = P(~(R#n) < ~(P) + e) ~ 1

because P(R~ C 7~O,q) = 1. Thus P~(An) -~ 1 for bc = per, and the same result follows for be = free because

suplHn(c~176 ~ v2 ~ ~i-Y=~ r i ~ L n _ r , j ~ L n

where v = supNc(Fq). This proves the proposition.

The proof of Theorem 2 is now completed as follows. First, there is no loss in assuming fi = 1. Next, we apply Propositions 4.1 and 5.4 to Fz = - p l o g z. Since p(Rn) = Nn/vn, this proves (2.22). Combining (2.22) with Proposition 4.1 (applied to F + Fz instead of F ) we arrive at the upper bound (2.20), and the lower bound (2.21) follows in the same way from (2.22) and Proposition 5.4.

6 Uniform estimates for tempered boundary conditions

This section is devoted to the proof of Theorem 3. We thus assume that ~o is regular and non-integrably divergent at the origin. By Proposition 3.2.8 of [17], the stable part ~pS of (p may be chosen bounded. So we can assume that the repulsive part ~o r satisfies (2.26). This gives us the following sharpening of the basic inequality (3.2).

Lemma 6.1 There exists a constant b < oc and an increasing function h : Z+ --+ [0, oc[ such that h(0) = 0,h(f)/• 2 --+ ec as ~ --+ oc, and for all n>O

Hn> T2 - bNn, (6.1)

188 H.-O. Georgii

where Thn = ~ h(Nc+i). (6.2)

iELn

Proof This is essentially Lemma 1 of [3]. For completeness we sketch the argument. Since the stable part o f q~ satisfies (2.2), and in view of the remarks above, we can assume without loss that ~o = ~o r >0 . We can further assume that )~(1) = 0 for the function X satisfying (2.26) and (2.27). It is also sufficient to prove (6.1) for n = 1.

Let co C C be a configuration o f some cardinality N > 3 d and K > 3 the integer part of N lid. Using (2,26) and writing Xk = Z(1/k) we then have

K

~o(x) > ~ ( ) ~ x - Zk_l)l{ixl __<x/h} k=2

for all x. For each k > 2 we divide C into k d cubes V(i) of size 1/k, and we let Ni = card co N V(i), i = 1,.. . , k d. Then

1 Z

x,yCa~,xOe y

k d

l{,x-yl<=l/k}~=~-~Q Ni )

i=1

1 > - ( k - d N 2 -- N) . = 2

Summing over k we thus obtain

1 ~ ( Z k - Zk-1)( k-dN2 - N) H1 (co) > k=2

K - 1

k=2

1 1 >N2d3-a= f Z ( s )J - lds =- h(N). - - 2 1 / ( K _ I )

Setting h(N) = 0 for N < 3 a we obtain the lemma. !

The next lemma establishes a simple relation between superquadratic and sub- quadratic functions on Z+ by means o f the Legendre-Fenchel transform. Observe that this transform preserves the parabola E -+ g2/2.

Lemma 6.2 Suppose g : Z+ --~ [0, ec[ is such that 9(0) = 0 and 9(E)ff 2 ~ ec as ( ---+ oe. Let g* : Z+ ~ [0, oc[ be defined by g*(m) = supt[mE - g(~)]. Then g* is increasing, g*(0) = O, and 9*(m)/m 2 ---+ 0 as m ---+ oc.

Proof This is a straightforward computation. Note that no convexity of 9 is required. 5!

The following lemma will allow us to reduce the case o f tempered boundary conditions to that of the free boundary condition.

Lemma 6.3 Let e > O, t > 0 be given. I f n is sufficiently large,


inf Hn,r > H. - aT h - ev., (r

where h is as in Lemma 6.1.

Proof In view o f (2.25), (2.5) and (4.3) we must show that

sup Z ~i-jXc+iNc+j(O < eT h + ev, (6.3) (E~2(t) ieLn,jf[Ln

for large n. Let q be such that h(E)_->f 2 for all f > q . Then T~<q2v, + Thn for all n. By Lemma 4.2, the contribution of all i,j with ]i - j [ _->k to the sum on the left-hand side of (6.3) is not larger than 6k[Th~ + vn(t2 d + q2)]. Choosing k suff• large we can achieve that this is less than e(T h + vn)/2.

To estimate the remaining part of the sum in (6.3) we define g ( Y ) = Eh(f) 1/2, f > 0. g is increasing with g ( 0 ) = 0 and satisfies g(E)/~ 2 --~ c~ and g(E)/h(E) ~ 0 as # ~ (x~. Let g* be as in Lemma 6.2. Then (m<=g(E) + g*(m) for all ( , m > 0 and therefore

~i-jNc+iNc+j(() iELn,jq{Ln:[i--j[ <k

<tP Z g(Nc+i)+ ~ Z g*(Nc+j(()) (6.4) iCLn\Ln-k jCLn+k\Ln

for all n > k and ~ E 12. Here 7 j = ~ i e z ~/" Since g ( f ) / h ( f ) ~ 0 as # ~ c~, the first expression on the right-hand side of (6.4) is at most eTh/2 + O(vn- Vn-k). Similarly, Lemma 6.2 implies that the second term on the right-hand side of (6.4) admits a bound of the form

eTn+k(O/4t + O(vn+k -- vn).

Since v n + k - v, = o(v~), it follows that, for ( E f2(t) and sufficiently large n, the left-hand side of (6.4) is not larger than e(T h + Vn)/2. This proves (6.3).

Proof of Theorem 3, assertion (a). Let F and t be as in the theorem, and let 0 < e < 1/2 be given. In view of Lemma 6.3 we have for sufficiently large n

fin(t) =- sup Qn (exp[-vnF(Rn) - H~,~])

<e'V"Q, (expt-vnF(Rn) - Hn + aTh~]) .

By Lemma 6.1 and the hypothesis on F , the exponent in the last integral is not larger than ev, + (c + b)N, - (1 - e)T h. An analogue of Lemma 4.5 thus shows that we may restrict the integral to a set o f the form {T~ h <sv~} with suitable s < oo, the remainder being at most e -~v" for any prescribed r > 0. Together with Proposition 4.1 ( for bc = free ), this gives the estimate

lim sup v~llogfin(t)<(a + es - 7) V (-r) ,

where 7 = in f [ I + ~ + Fist]. Letting first e--+ 0 and then z ~ ~ we obtain the uniform upper bound

190 H.-O. Georgii

lim sup v] -1 logfin( t )< - -y . (6.5) n----+ OO

To obtain a lower estimate o f

P--n(t)------ ~c~(t)inf Qn (exp[-vnF(Rn)-Hn,~])

we can proceed just as in the proof of Proposition 5.4, the only difference being that now we must show that

infPn_r(v-fflHn,~ < (b (P )+~) --+ 1 as n - - + o o ~o(t)

for any given e > 0, each ergodic P E 7~o which is supported on some I'q, and suitable r > r(cp). But (2.7), (4.3) and Lemma 4.2 imply that

sup Hn,r <=H. + e vn/2 on Fq f"l Qn-r

when r is large enough, and we know from the proof of Proposition 5.4 that

Pn_r(v21H, < O(P) + e/2) --+ 1 as n ---, oo.

We thus arrive at the uniform lower bound

l i m inf v~ -1 logp_n(t ) > - inf [ I + @ + FUSC]. (6.6)

Assertion (a) of Theorem 3 now follows from (6.5) and (6.6) in the same way as Theorem 2 from Propositions 4.1 and 5.4. '

We now turn to the large deviation principle for tempered Gibbs measures. Our main tool are the remarkable probability estimates of Ruelle [18]. (Similar estimates appear in [3].) The implication ( a ) ~ ( b ) of his Corollary 5.3 gives us the following.

Proposit ion 6.4 For given z, fl > O, there exist constants 7, fi > 0 such that, for all tempered Gibbs measures P with parameters z, fi and all n > O, Pn is absolutely continuous relative to Q, with a density fn satisfying f~ <exp[v~ - yT~ + 6N,].

For t > 0 and n > 0 we define

~2,(t) = {~ E (2" ~ \ An E f2(t)}. (6.7)

Corol lary 6.5 Let z, fl > 0 and ~,c > 0 be any constants. Then there exists some t > 0 such that. for each tempered Gibbs measure P with parameters z, t3 and all n>O,

p(ecXn; ~n(t) c) <e -T~

Proof We may assume that ~ > log 2. Let 7, 3 be as in Proposition 6.4 and t so large that t y - e~+a> ~. Then we can write

Large deviations for Gibbsian particle systems

p(ecNn;Qn(t)c)<= Zp(ecN";T~ > tv~) f > n

< Z P(exp[cNz + yTl - ytvt])

191

<= Z Q(exp[vt(c + 6)Nt - 7tvt]) f > n

< Z e x p [ - w t ] . f > n

This implies the corollary because v~ > v, + ~ - n when f > n.

Proof o f Theorem 3, assertion (b). Let z, fi > 0 and P be a tempered Gibbs measure for z, fl. Also, let F : 79o ---* R U {oo} be such that F > - e(1 + p ) for some c < oc, T > c an arbitrary constant, and t as in Corollary 6.5. Then

P (eVnF(R"); f2n (t) c) <= e -(~-~)~"

and P(eV"F(R"); O.(t)) = fa.(oP(d~)P.,z3,r (e v"r(R"))

=< sup P.,~3,r v"r(R"))

for all n. The uniform upper bound (2.28) thus implies

lira sup v~ -1 logP(e v'F(R~)) < - inf[/~ 3 + F1s,] A (z - e ), n--+oo

and letting z --* oo we obtain the upper large deviation bound for P. The lower bound follows from the inequality

P(e vnF(R")) >=P(f2(t)) inf P. z,~ ~(ev"F(R")) ~c~(t) ' -'

together with (2.29) and the fact that P(~2(t)) > 0 for sufficiently large t. L

7 The equivalence of ensembles

In this section we prove Theorem 4. let (p be superstable and regular. We first look at the function q~ defined in (2.31). Let

f 2 ~ = { c o E f ~ ' ~ o ( x - y ) < oo fo ra l l x, y E o ) , x 4 : y }

and

v((p) = sup{p (P) 'P E Po,I(P) < ec,P(f2~o) = 1}.- (7.1)

Clearly, v(~o) depends on q~ only via the set {~o = co}. I f q) is finite (except possibly at the origin) then v(~o) = (x). For in this case we have f2e = O so that the Poisson point random fields P = Q~ of arbitrarily large intensity z appear on the right-hand side of (7.1).

L e m m a 7.1 q5 is convex. On [0,v(cp)[, ~b isfinite and continuous. Also, 4~(v)>av 2 - by for all v>O and the constants a,b in (3.3), and if 2 = f ~o(x)dx is finite then ~b(v) =< 2v2/2 for all v >= O. (In particular, it follows that 2 > a.)

192 H.-O. Georgii

Proof The inequalities for 05 follow from (3.3) and the easily verified fact that q}(QZ) = )bz2/2 for all z > 0. It only remains to show that 05 is finite on [0, v(~0)[. For, together with the obvious convexity of 05, this will imply that 05 is continuous on ]0, v(~o)[. The continuity o f 05 at 0 is clear because 05(0) = 0 and 05(v) > - by.

To prove the finiteness o f 05 we fix any v < v(~0). By (7.1) there exists some P E 7)o such that p(P) > v,I(P) < c~, and P(f2~) = 1. A glance at the proof of Lemma 5.1 shows that its hypothesis ~(P) < cc can be replaced by the condition P(~2~) = 1. Therefore we can find some U E 7~o such that p(P') > v,I(U) < oo, and pt(l'q) = 1 for some q < c~. Since Fq is O-invariant, the Palm measure of U is also supported on /'q. This and the regularity of ~o immediately imply that �9 (U) < co. Writing s = v/p(P') we thus obtain that P" =- sP' + (1 - s)6r has the properties p(P")= v, ~ ( P " ) < ec, and I(P") < ec. Hence 05(v) < c~.

In view of (7.3) and (7.7) below, the function 05(v) coincides with the function t0(p) on page 50 of [17]. Note, however, that not necessarily 05(v)-+ oc when v -+ v(qo) < c~; indeed, i f ~o is a pure hard core potential (taking only the values 0 and ~ ) then 05(v) = 0 for all v < v(cp). Next we define

s(v,e) = - i n f { I (P) : P E 7>o, p(P) = v, ~ ( P ) < e } . (7.2)

Clearly, s( . , - ) is increasing in e and concave, and (2.31) means that

05(v) = inf{s(v,.) > - c o } for all v > 0 . (7.3)

Using the lower semicontinuity of �9 and the fact that I has compact level sets [10] we also see that s ( . , . ) is upper semicontinuous, s(-, .) is the entropy density, as is shown in the next lemma which proves (2.33).

Lemma 7.2 Let D C [0,ec[ be a nondegenerate interval with i n f D < v(~o), and let e > inf05(D). Then

lira v~-llog Qn(Nn E vnD, Hn,per ~Vn8) = s(D, ~) --= sups(v, e). n---~oo vGD

I f infD > v(q~) or e < inf05(D) then the limit above equals -oc.

Proof L e t / 5 be the closure and D ~ the interior o f D. By the continuity of 05 on [0, v(q~)[, inf05(/5) = inf05(D ~ = inf05(D). In view of the continuity of p and the lower semicontinuity of ~b, the set A = {p E/5, 4~<e} is closed. The upper bound (2.20) for/~ = 0,z = 1 thus implies that

lim sup vn I log Qn(Rn C A)< - mini (A) (7.4) n----+ o o

(I attains its minimum over A because I has compact level sets.) On the other hand, consider the convex set U = { p E D ~ ,~b < ~}. Since e >inf05(D~ U M { I < ec} =t=~. A standard convexity argument (together with the fact that I , p and are affine) thus shows that inf I(U) = infI(A) = s(D, e). But

lim inf v~ -1 log Qn(R~ c U ) > - i n f I ( U ) n----+ OO

by the arguments of Sect. 5. Indeed, let P E U be such that I(P) < oo. For given e > 0, Lemma 5.1 provides us with some ergodic U E U such that I (U) < I(P) -4-


e and P~(Fq) = 1 for some q > 0. As in the proof of Proposition 5.4, we have the estimate

v2~logQn(Rn E U)>= - I (P ' ) - e + v211ogP'(R#~ C U, v2~logs ~ Z ( P ') -t- ~),

and McMillan's theorem [5, 14] and Lemmas 5.2 and 5.3 show that the last probability tends to 1 as n ~ o~. The second statement of the lemma follows from (7.3) and (7.4). _J

Proposition 7.3 Suppose D C [0, oo[ is a nondegenerate interval with infD < v(~o), and let e > infr Then the sequence (QnlD,~,pcr)n>=O is relatively compact (in vc) , and every accumulation point belongs to the set MD,~ o f all I-minimizers in {p ~ JO,~<=~}.

Proo f Lemma 7.2 implies that the conditioning event in (2.30) has positive probability for sufficiently large n. Thus, for all these n, QnlD,~,per is well-defined. Let

/3(n) E 79 denote the measure relative to which the configurations in the disjoint blocks An + (2n + 1)i, i E L, are independent with identical distribution QnlD,e,per,

and p(n) E 7)0 the invariant average of/3(n) o Oxl,X E An. Since QnlD,~,per is invariant

under translations modulo An, p(n)(f) _ QnlD,~,per(f) --+ 0 as n --+ oc, for all f E s cf. Lemma 4.6 of [8]. Just as in Step 3) of the proof of Lemma 5.1 we obtain

I (P (n)) < v2 t I(QnID,~,p~r; Qn) = - v2 ~ logQn (p(Rn) E D, ~b(Rn) < ~).

Lemma 7.2 thus implies that

lim supI(p(n))< - s ( D , e ) < co. n --+ o G

Since I has compact (and sequentially compact) level sets, it follows that the sequence (P(n))n > 0, and thus also the asymptotically equivalent sequence (QnID,~,per)n >=o, are relatively compact. Moreover, each accumulation point of any of these sequences belongs to the set { I < - s (D,e)} . On the other hand, Lemma 5.7 of [10] asserts that p(n) is asymptotically equivalent to

QnlD,e,perR n =~ f QnlD,~,per(dCo)Rn,o)

as n--~ ec, and the latter measures all belong to the closed convex set {p E /5, ~b =<e}, as is easily seen by approximating QnlD,~,pcr by suitable discrete measures. This proves the proposition, l

To deduce the first part of (2.32) from Proposition 7.3 we choose a tangent plane (v, e t) ---* p + fie' - vlogz to the concave function s(.,-) on the subset o f / } x ] - cx~, e] on which s(., .) attains its maximum s(D, e). The monotonicity of s(v, .) implies that t3 > 0. It then follows that, for all P E 7)owith 05(p) < cx~,

I (P) > - s(p(P), ~(P)) > - p - f l~(P) + p(P)logz

with equality when P E MD,~. Since MD,~ + ~, we may conclude that p = p(z, t ) and

ML,,~ c (Iz,~ = 0}. (7.5)

194 H.-O. Georgii

Together with Proposition 7.3, this gives the first part o f (2.32). The second part is the subject of the next proposition.

Proposition 7.4 For all z, fi > O, the sequence (Pn,z,fi,per)n>O is relatively compact, and all its accumulation points belong to {I~,~ = 0}.

P r o o f We proceed just as in the proof o f the last proposition. For each n > 0 we have

_ Vn 1 lOgZn,z,fi,per =V n 11(Pn,z,t3,pe r; Qn)

-}- flen,z,fi,per((1)(Rn)) - Pn,z,fl,per(V2 INn)logz.

The first term on the right-hand side is not less than I(p(n)), where p(n) is the in-

variant An-average o f the An-periodic measure / 3(n) making the configurations in the blocks An + (2n + l)i, i E L, independent with identical distribution Pn,z,~,per. Using

Lemma 3.3 and the observation that Pn,z,~,per(p(2)(R.)) = p(2)(pn,z,p,per) < oc we see further that the second term on the right-hand side is not less than fl~(Pn,z&perRn). Combining this with (2.22) we obtain

lira sup [i(p(,)) + ficD(Pn,z,fl,perRn ) _ p(p(n))logz] _<_ _ p(z, fi) < oc. / 7 - - - + 0 0

The compactness of the level sets of I , the lower semicontinuity of ~, and the asymptotic equivalence of P(n),P,,z&pcrRn, and Pn,z,3,per thus give the result.

To conclude, we mention without proof that the entropy function s(., .) is related to the pressure by the classical Legendre transformation as follows. For all z, ~ > 0,

p(z, 3) = max - / ? 8 + vlogz].

Conversely, s(v, e) = min [p(z, ~) + fie - vlogz]

fi>0,z>0

whenever v > 0 and s(v, e) > - o c .

(7.6)

(7.7)

Acknowledgement. I would like to thank Y. Suhov for bringing reference [6] to my attention, and an anonymous referee for reminding me of reference [3].

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[8] Georgii, H.O.: Large deviations and maximum entropy principle for interacting random fields on Z a. Ann. Probab. 21, 1845-1875 (1993)

[9] Georgii, H.O.: Large deviations for hard core particle systems. In: Kotecky, R. (ed.) Pro- ceedings of the I992 Prague Workshop on Phase Transitions. Singapore: World Scientific 1993

[10] Georgii, H.O., Zessin, H.: Large deviations and the maximum entropy principle for marked point random fields. Probab. Theory Relat. Fields 96, 177-204 (1993)

[11] Lanford, O.E.: Entropy and equilibrium states in classical statistical mechanics. In: Lenard, A. (ed.) Statistical Mechanics and Mathematical Problems. (Lect. Notes Phys. 20) Berlin Heidelberg New York: Springer 1973

[12] Martin-L6f, A.: Statistical Mechanics and the Foundations of Thermodynamics. (Lect. Notes Phys. 101) Berlin Heidelberg New York: Springer 1979

[13] Matthes, K., Kerstan, J., Mecke, J.: Infinitely Divisible Point Processes. Chichester: Wiley 1978

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[15] Olla, S., Varadhan, S.R.S., Yau, H.T.: Hydrodynamic limit for a Hamiltonian system with a weak noise, Commun. Math. Phys. 155, 523-560 (1993)

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